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Articles 1 - 30 of 58
Full-Text Articles in Algebra
Counting Hamming-Graceful Labelings Of Paths, Ashka Dalal
Counting Hamming-Graceful Labelings Of Paths, Ashka Dalal
Mathematical Sciences Technical Reports (MSTR)
Let Γ be a graph of m edges and n vertices. A Hamming-graceful labeling of Γ labels vertices with binary strings of length m and the edge labels are induced by the Hamming distance between vertex labels. It is known that all paths have Hamming-graceful labelings, thus the question arises, how many possible labelings exist for a path of a given size. We develop an algebraic way to generate labelings, conjecture a method for counting, prove this for small examples, and verify larger examples using a Python program.
Branching Matrices For The Automorphism Group Lattice Of A Riemann Surface, Sean A. Broughton
Branching Matrices For The Automorphism Group Lattice Of A Riemann Surface, Sean A. Broughton
Mathematical Sciences Technical Reports (MSTR)
Let S be a Riemann surface and G a large subgroup of Aut(S) (Aut(S) may be unknown). We are particularly interested in regular n-gonal surfaces, i.e., the quotient surface S/G (and hence S/Aut(S)) has genus zero. For various H the ramification information of the branched coverings S/K -> S/H may be captured in a matrix. The ramification information, in particular strong branching, may be then be used in analyzing the structure of Aut(S). The ramification information is conjugation invariant so the matrix's rows and columns may be indexed by conjugacy classes of subgroups. The only required …
Topological And Hq Equivalence Of Prime Cyclic P-Gonal Actions On Riemann Surfaces (Corrected), Sean A. Broughton
Topological And Hq Equivalence Of Prime Cyclic P-Gonal Actions On Riemann Surfaces (Corrected), Sean A. Broughton
Mathematical Sciences Technical Reports (MSTR)
Two Riemann surfaces S1 and S2 with conformal G-actions have topologically equivalent actions if there is a homeomorphism h : S1 -> S2 which intertwines the actions. A weaker equivalence may be defined by comparing the representations of G on the spaces of holomorphic q-differentials Hq(S1) and Hq(S2). In this note we study the differences between topological equivalence and Hq equivalence of prime cyclic actions, where S1/G and S2/G have genus zero.
Continuous Dependence Of Solutions Of Equations On Parameters, Sean A. Broughton
Continuous Dependence Of Solutions Of Equations On Parameters, Sean A. Broughton
Mathematical Sciences Technical Reports (MSTR)
It is shown under very general conditions that the solutions of equations depend continuously on the coefficients or parameters of the equations. The standard examples are solutions of monic polynomial equations and the eigenvalues of a matrix. However, the proof methods apply to any finite map T : Cn -> Cn.
Calculation Of The Killing Form Of A Simple Lie Group, Sean A. Broughton
Calculation Of The Killing Form Of A Simple Lie Group, Sean A. Broughton
Mathematical Sciences Technical Reports (MSTR)
The Killing form of a simple Lie Algebra is determined from invariants of the extended root diagrams of the Lie algebra.
The Square Discrete Exponentiation Map, A Wood
The Square Discrete Exponentiation Map, A Wood
Mathematical Sciences Technical Reports (MSTR)
We will examine the square discrete exponentiation map and its properties. The square discrete exponentiation map is a variation on a commonly seen problem in cryptographic algorithms. This paper focuses on understanding the underlying structure of the functional graphs generated by this map. Specifically, this paper focuses on explaining the in-degree of graphs of safe primes, which are primes of the form p = 2q + 1, where q is also prime.
Algebraic Solutions To Overdefined Systems With Applications To Cryptanalysis, Eric Crockett
Algebraic Solutions To Overdefined Systems With Applications To Cryptanalysis, Eric Crockett
Mathematical Sciences Technical Reports (MSTR)
Cryptographic algorithms are based on a wide variety of difficult problems in mathematics. One of these problems is finding a solution to a system of multivariate quadratic equations (MQ). A generalization of this problem is to find a solution to a system of higher order non-linear equations. Both of these problems are NP-hard over any field. Many cryptosystems such as AES, Serpent, Toyocrypt, and others can be reduced to some form of the MQ problem. In this paper we analyze the relinearization and XL algorithms for solving overdetermined systems of non-linear equations, as well as two variations of the XL …
The Digraph Of The Square Mapping On Elliptic Curves, Katrina Glaeser
The Digraph Of The Square Mapping On Elliptic Curves, Katrina Glaeser
Mathematical Sciences Technical Reports (MSTR)
Consider a subgroup of an elliptic curve generated by a point P of order n. It is possible to match any point Q to an integer k (mod n) such that Q = kP using a brute force method. By observing patterns in the digraph of the squaring map on the integers modulo n it is possible to perform this matching. These techniques can be applied to solving the Elliptic Curve Discrete Log Problem given a complete graph of the square mapping k P -> k^2 P for the elliptic curve points.
Discrete Logarithm Over Composite Moduli, Marcus L. Mace
Discrete Logarithm Over Composite Moduli, Marcus L. Mace
Mathematical Sciences Technical Reports (MSTR)
In an age of digital information, security is of utmost importance. Many encryption schemes, such as the Diffie-Hellman Key Agreement and RSA Cryptosystem, use a function which maps x to y by a modular power map with generator g. The inverse of this function - trying to find x from y - is called the discrete logarithm problem. In most cases, n is a prime number. In some cases, however, n may be a composite number. In particular, we will look at when n = p^b for a prime p. We will show different techniques of obtaining graphs of this …
Structural Properties Of Power Digraphs Modulo N, Joseph Kramer-Miller
Structural Properties Of Power Digraphs Modulo N, Joseph Kramer-Miller
Mathematical Sciences Technical Reports (MSTR)
We define G(n, k) to be a directed graph whose set of vertices is {0, 1, ..., n−1} and whose set of edges is defined by a modular relation. We say that G(n, k) is symmetric of order m if we can partition G(n, k) into subgraphs, each containing m components, such that all the components in a subgraph are isomorphic. We develop necessary and sufficient conditions for G(n, k) to contain symmetry when n is odd and square-free. Additionally, we use group theory to describe the structural properties of the subgraph of G(n, k) containing only those vertices relatively …
The Barycenter Of The Numerical Range Of A Matrix, Sean A. Broughton, Roger G. Lautzenheiser, Thomas Werne
The Barycenter Of The Numerical Range Of A Matrix, Sean A. Broughton, Roger G. Lautzenheiser, Thomas Werne
Mathematical Sciences Technical Reports (MSTR)
The numerical range W(A) of an nxn matrix A is the totality of the scalar products <Ax,x> as x varies over all unit vectors in Cn The barycenter (center of mass) of the numerical range is defined geometrically as the center of mass of W(A) considered as a planar lamina with variable density and also as a limit of sample averages (<Ax1,x1>+...+<AxN,xN>)/N. Under a wide range the sampling schemes it is shown that the barycenter is the average of the spectrum …
Isomorphisms Of Elliptic Curves Over Extensions Of Finite Fields, Mathew Niemerg
Isomorphisms Of Elliptic Curves Over Extensions Of Finite Fields, Mathew Niemerg
Mathematical Sciences Technical Reports (MSTR)
Our main interest lies in exploring isomorphisms of elliptic curves. In particular, we focus on two curves defined over a base field and look at which extension fields the curves are isomorphic over. Elliptic curves have a fascinating structure behind them. This structure allows for much to be explored and studied.
On The Action Of Weight-Preserving Sets, Matthew Badger
On The Action Of Weight-Preserving Sets, Matthew Badger
Mathematical Sciences Technical Reports (MSTR)
We introduce weight-preserving sets of binary words. Any transformation that respects the row and column weights of a 0-1 matrix can be decomposed as a composition of two types of action on the matrix. We conjecture that weight-preserving sets perform only one type of action, permutations of rows and columns; i.e., weight-preserving sets are cwatsets.
Big Cwatsets And Hamming Code, Matthew Davis, Thomas M. Langley, Norah Mazel
Big Cwatsets And Hamming Code, Matthew Davis, Thomas M. Langley, Norah Mazel
Mathematical Sciences Technical Reports (MSTR)
In contrast to Lagrange's Theorem in Finite Group Theory, we show that the ratio of the largest proper cwatset of degree d to the size of binary d-space approaches 1 as d approaches infinity. We show how to explicitly construct large cwatsets as cosets of Hamming Codes, and discuss many open questions that arise.
The Galois Correspondence For Branched Covering Spaces And Its Relationship To Hecke Algebras, Matthew Ong
The Galois Correspondence For Branched Covering Spaces And Its Relationship To Hecke Algebras, Matthew Ong
Mathematical Sciences Technical Reports (MSTR)
There is a very beautiful correspondence between branched covers of the Riemann sphere P1 and subgroups of the fundamental group π1(P1 − {branch points}), exactly analogous to the correspondence between subfields of an algebraic extension E/F and subgroups of the Galois group Gal(E/F). This paper explores the concept of a Hecke algebra, which in this context is a generalization of the Galois group to the case of non- Galois covers S/P1. Specifically, we show that the isomorphism type of a Hecke algebra C[H\G/H] is completely determined by the decomposition of …
Tilings Of Low-Genus Surfaces By Quadrilaterals, John Gregoire, Isabel Averil
Tilings Of Low-Genus Surfaces By Quadrilaterals, John Gregoire, Isabel Averil
Mathematical Sciences Technical Reports (MSTR)
In contribution to the classification of all tilings of low-genus surfaces, the kaleidoscopic and non-kaleidoscopic tilings by quadrilaterals are given up to genus 12. As part of their classification, the algebraic structure of the conformal tiling groups and the geometric structure of the tiles are specified. In addition, several infinite classes of tilings and tiling groups are presented.
Triangular Surface Tiling Groups For Low Genus, Sean A. Broughton, Robert M. Dirks, Maria Sloughter, C. Ryan Vinroot
Triangular Surface Tiling Groups For Low Genus, Sean A. Broughton, Robert M. Dirks, Maria Sloughter, C. Ryan Vinroot
Mathematical Sciences Technical Reports (MSTR)
Consider a surface, S, with a kaleidoscopic tiling by non-obtuse triangles (tiles), i.e., each local reflection in a side of a triangle extends to an isometry of the surface, preserving the tiling. The tiling is geodesic if the side of each triangle extends to a closed geodesic on the surface consisting of edges of tiles. The reflection group G*, generated by these reflections, is called the tiling group of the surface. This paper classifies, up to isometry, all geodesic, kaleidoscopic tilings by triangles, of hyperbolic surfaces of genus up to 13. As a part of this classification the tiling groups …
Classification Of Cwatsets Through Order 23, Ben Goodwin, Dennis Lin
Classification Of Cwatsets Through Order 23, Ben Goodwin, Dennis Lin
Mathematical Sciences Technical Reports (MSTR)
A cwatset of order n can be represented by a transitive subgroup of Sn. Previous work has shown that each conjugacy class of representation groups corresponds to an isomorphism class of cwatsets. We present a technique for determining whether a particular transitive subgroup of Sn can appear as the representation group for a cwatset of order n. Using this method, we provide a full classification of cwatset isomorphism classes through order 23.
Quest For Tilings On Riemann Surfaces Of Genus Six And Seven, Robert Dirks, Maria Sloughter
Quest For Tilings On Riemann Surfaces Of Genus Six And Seven, Robert Dirks, Maria Sloughter
Mathematical Sciences Technical Reports (MSTR)
The problem of kaleidoscopically tiling a surface by congruent triangles is equivalent to finding groups generated in certain ways. In order to admit a tiling, a group must have a specific set of generators as well as an involutary automorphism, T, that acts to reverse the orientation of the tiles. The purpose of this paper is to explore group theoretic and computational methods for determining the existence of symmetry groups and tiling groups, as well as to classify the symmetry and tiling groups on hyperbolic Riemann surfaces of genus 6 and 7.
Cwatset Isomorphism And Its Consequences, Carolyn M. Girod, Matthew Lipinski, Joseph R. Mileti, Jennifer R. Paulhus
Cwatset Isomorphism And Its Consequences, Carolyn M. Girod, Matthew Lipinski, Joseph R. Mileti, Jennifer R. Paulhus
Mathematical Sciences Technical Reports (MSTR)
We explore the consequences of cwatset isomorphism (there are a finite number of non-isomorphic cwatsets of each order) and consider parallels between the theory of groups and the theory of cwatsets (cwatsets of prime order are cyclic but direct sums of isomorphic cwatsets aren't necessarily isomorphic).
Splitting Tiled Surfaces With Abelian Conformal Tiling Group, Sean A. Broughton
Splitting Tiled Surfaces With Abelian Conformal Tiling Group, Sean A. Broughton
Mathematical Sciences Technical Reports (MSTR)
Let p be a reflection on a closed Riemann Surface S, i.e., an anti-conformal involutary isometry of S with a non-empty fixed point subset. Let Sp denote the fixed point subset of p, which is also called the mirror of p. If S −Sp has two components, then p is called separating and we say that S splits at the mirror Sp. Otherwise p is called non-separating. We assume that the system of mirrors, Sq, as q varies over all reflections in the isometry group Aut*(S) defines a tiling of the surface, consisting of triangles. In turn, the tiling determines …
Divisible Tilings In The Hyperbolic Plane, Sean A. Broughton, Dawn M. Haney, Lori T. Mckeough, Brandy M. Smith
Divisible Tilings In The Hyperbolic Plane, Sean A. Broughton, Dawn M. Haney, Lori T. Mckeough, Brandy M. Smith
Mathematical Sciences Technical Reports (MSTR)
We consider triangle-quadrilateral pairs in the hyperbolic plane which "kaleidoscopically" tile the plane simultaneously. In this case the tiling by quadrilaterals is called a divisible tiling. All possible such divisible tilings are classified. There are a finite number of 1,2, and 3 parameter families as well as a finite number of exceptional cases.
Tilings Which Split A Mirror, Jim Belk
Tilings Which Split A Mirror, Jim Belk
Mathematical Sciences Technical Reports (MSTR)
We consider the mirror of a reflection which consists of its subset of fixed points. We investigate a number of conditions on the tiling that guarantee that the surface splits at a mirror.
Automorphic Subsets Of The N-Dimensional Cube Are Translations Of Cwatsets, Matthew Lepinski
Automorphic Subsets Of The N-Dimensional Cube Are Translations Of Cwatsets, Matthew Lepinski
Mathematical Sciences Technical Reports (MSTR)
It is known that automorphic subsets are generalizations of cwatsets. In this paper we show that an automorphic subset is the translation of some cwatset, and therefore that each automorphic subset is internally isomorphic to a cwatset.
Symmetry And Tiling Groups For Genus 4 And 5, C. Ryan Vinroot
Symmetry And Tiling Groups For Genus 4 And 5, C. Ryan Vinroot
Mathematical Sciences Technical Reports (MSTR)
All symmetry groups for surfaces of genus 2 and 3 are known. In this paper, we classify symmetry groups and tiling groups with three branch points for surfaces of genus 4 and 5. Also, a class of symmetry groups that are not tiling groups is presented, as well as a class of odd order non-abelian tiling groups.
Quadrilaterals Subdivided By Triangles In The Hyperbolic Plane, Dawn M. Haney, Lori T. Mckeough
Quadrilaterals Subdivided By Triangles In The Hyperbolic Plane, Dawn M. Haney, Lori T. Mckeough
Mathematical Sciences Technical Reports (MSTR)
In this paper, we consider triangle-quadrilateral pairs in the hyperbolic plane which “kaleidoscopically” tile the plane simultaneously. These tilings are called divisible tilings or subdivided tilings. We restrict our attention to the simplest case of divisible tilings, satisfying the corner condition, in which a single triangle occurs at each vertexof the quadrilateral. All possible such divisible tilings are catalogued as well as determining the minimal genus surface on which the divisible tiling exists. The tiling groups of these surfaces are also determined.
Generalized Conjugacy Classes, Pramod N. Achar
Generalized Conjugacy Classes, Pramod N. Achar
Mathematical Sciences Technical Reports (MSTR)
Generalized conjugation is the action of a group on its underlying set given by (g,x) -> p(g)xg-1, where p is some fixed endomorphism of G. Here we study combinatorial properties of the sizes of the orbits of the preceding action. In particular, we reduce the problem to a simpler case if p has nontrivial kernel, or if it is an inner automorphism, and we give a construction that allows a partial analysis in the general case.
Cwatsets: Weights, Cardinalities, And Generalizations, Richard Mohr
Cwatsets: Weights, Cardinalities, And Generalizations, Richard Mohr
Mathematical Sciences Technical Reports (MSTR)
This report provides an upper bound on the average weight of an element in a cwatset and discusses the ratio of the cardinality of a cwatset to the cardinality of the group containing the cwatset. The concept of a generalized cwatset is also introduced.
Rectangular Groups, Nick Fiala, Crystal Hanscom, Patrick Keenan, Tung Tran
Rectangular Groups, Nick Fiala, Crystal Hanscom, Patrick Keenan, Tung Tran
Mathematical Sciences Technical Reports (MSTR)
We provide an overview of results and conjectures relating to rectangular groups.
Conjugacy Classes Of Triple Products In Finite Groups, Kevin Hutson, Emily Salvo
Conjugacy Classes Of Triple Products In Finite Groups, Kevin Hutson, Emily Salvo
Mathematical Sciences Technical Reports (MSTR)
For an underlying finite group G, we establish estimates on the number of triples that bind a certain set to one conjugacy class, or else breaks it into two conjugacy classes.